Sure, let's simplify the expression step-by-step and classify the resulting polynomial.
The given expression is:
[tex]\[
4x(x + 1) - (3x - 8)(x + 4)
\][/tex]
First, let's expand the terms separately:
For [tex]\(4x(x + 1)\)[/tex]:
[tex]\[
4x(x + 1) = 4x^2 + 4x
\][/tex]
For [tex]\((3x - 8)(x + 4)\)[/tex], we use the distributive property (FOIL method):
[tex]\[
(3x - 8)(x + 4) = 3x(x) + 3x(4) - 8(x) - 8(4)
\][/tex]
[tex]\[
= 3x^2 + 12x - 8x - 32
\][/tex]
[tex]\[
= 3x^2 + 4x - 32
\][/tex]
Now, we substitute these expanded expressions back into the original expression:
[tex]\[
4x(x + 1) - (3x - 8)(x + 4) = (4x^2 + 4x) - (3x^2 + 4x - 32)
\][/tex]
Next, distribute the negative sign in the second term:
[tex]\[
= 4x^2 + 4x - 3x^2 - 4x + 32
\][/tex]
Combine like terms:
[tex]\[
= (4x^2 - 3x^2) + (4x - 4x) + 32
\][/tex]
[tex]\[
= x^2 + 32
\][/tex]
We now have the simplified expression:
[tex]\[
x^2 + 32
\][/tex]
To classify the polynomial, observe:
1. The highest degree term is [tex]\(x^2\)[/tex], which indicates that it is a quadratic polynomial.
2. There are two terms: [tex]\(x^2\)[/tex] and [tex]\(32\)[/tex].
Therefore, the simplified expression [tex]\(x^2 + 32\)[/tex] is a quadratic polynomial with two terms.
So, the classification of the resulting polynomial is:
[tex]\[
\boxed{D. \text{quadratic binomial}}
\][/tex]
